Multi-Value Numerical Modeling for Special Di erential Problems

Abstract

The subject of this thesis is the analysis and development of new numerical methods for Ordinary Di erential Equations (ODEs). This studies are motivated by the fundamental role that ODEs play in applied mathematics and applied sciences in general. In particular, as is well known, ODEs are successfully used to describe phenomena evolving in time, but it is often very di cult or even impossible to nd a solution in closed form, since a general formula for the exact solution has never been found, apart from special cases. The most important cases in the applications are systems of ODEs, whose exact solution is even harder to nd; then the role played by numerical integrators for ODEs is fundamental to many applied scientists. It is probably impossible to count all the scienti c papers that made use of numerical integrators during the last century and this is enough to recognize the importance of them in the progress of modern science. Moreover, in modern research, models keep getting more complicated, in order to catch more and more peculiarities of the physical systems they describe, thus it is crucial to keep improving numerical integrator's e ciency and accuracy. The rst, simpler and most famous numerical integrator was introduced by Euler in 1768 and it is nowadays still used very often in many situations, especially in educational settings because of its immediacy, but also in the practical integration of simple and well-behaved systems of ODEs. Since that time, many mathematicians and applied scientists devoted their time to the research of new and more e cient methods (in terms of accuracy and computational cost). The development of numerical integrators followed both the scienti c interests and the technological progress of the ages during whom they were developed. In XIX century, when most of the calculations were executed by hand or at most with mechanical calculators, Adams and Bashfort introduced the rst linear multistep methods (1855) and the rst Runge- Kutta methods appeared (1895-1905) due to the early works of Carl Runge and Martin Kutta. Both multistep and Runge-Kutta methods generated an incredible amount of research and of great results, providing a great understanding of them and making them very reliable in the numerical integration of a large number of practical problems. It was only with the advent of the rst electronic computers that the computational cost started to be a less crucial problem and the research e orts started to move towards the development of problem-oriented methods. It is probably possible to say that the rst class of problems that needed an ad-hoc numerical treatment was that of sti problems. These problems require highly stable numerical integrators (see Section ??) or, in the worst cases, a reformulation of the problem itself. Crucial contributions to the theory of numerical integrators for ODEs were given in the XX century by J.C. Butcher, who developed a theory of order for Runge-Kutta methods based on rooted trees and introduced the family of General Linear Methods together with K. Burrage, that uni ed all the known families of methods for rst order ODEs under a single formulation. General Linear Methods are multistagemultivalue methods that combine the characteristics of Runge-Kutta and Linear Multistep integrators... [edited by Author]